Loading....
Coupon Accepted Successfully!

 

Number of Exponents

Let us take a simple number – 105
 
This is read as – 10 to the power 5, or we say that exponent of 10 is 5.
 
In simple terms, exponents are also known as Power.
 
Example-1
What is the maximum value of s if N = (35 × 45 × 55 × 60 × 124 × 75) is divisible by 5s?
Solution
If we factorize N = (35 × 45 × 55 × 60 × 124 × 75), then we can see that 5 appears 6 times, it means N is divisible by 56.
So, maximum value of s = 6
Exponent of any prime number P in n! = Description: 5679.png, where n ≥ px and [.] denotes the greatest integer value i.e., we have to consider only the integral value.
 
Let us find the exponent of 5 in 1000! = 1000/5 + 1000/52 + 1000/53 + 1000/54 = 200 + 40 + 8 + 1 = 249
 
 
Example-2
What is the highest power of 5 which can divide N = (22! + 17894!)?
Solution
Number of times this number is divisible by 5 is same as the number of zeroes at the end of this number. Since 22! have 4 zeroes at its end, so N will also be having only four zeroes at its end. Hence, highest power of 5, which can divide N is 4.
 

Process to find out the exponent of any composite number in n!

There are three different kinds of composite numbers.
  1. Product of two or more than two prime numbers with unit power of all the prime numbers
     
    For example, 15 (5 × 3), 30 (2 × 3 × 5) etc.
  2. (Any prime number)n, where n >1
     
    For example, 4 (22), 27 (33)
  3. Product of two or more than two prime numbers with power of any one prime number more than 1.
     
    For example, 12 (22 × 3), 72 (23 × 32) etc.
Let us find out the exponents of the above-written composite numbers one by one:
  • Find out the exponent of 15 in 100!
    15 is the product of two distinct prime numbers 5 and 3.
So, to find out the exponents of 15, we need to find out the exponents of 5 and 3 individually.
 
So, by applying the same formula of finding out the exponents for any prime number in both of these cases individually, minimum of those two will be the answer.
100/5x = [100/5] + [100/52] = 20 + 4 = 24
100/3x = [100/3] + [100/32] + [100/33] + [100/34
= 33 + 11 + 3 + 1 = 48
Obviously, 24 is going to be the answer.
  • Find out the exponent of 25 in 100
    25 = 52
    In this case, we first find out the exponents of 5 and then divide it by 2 (actually the power) to find the exponents of 25.
    100/5x = [100/5] + [100/52] = 20 + 4 = 24
    So, 100/25x = 24/2 = 12
  • Similarly, we can find out for the third category of numbers also.




Test Your Skills Now!
Take a Quiz now
Reviewer Name